The birthplaces of martial arts around the world were various militaries, policing organisations, and sports. Militaries need to train their soldiers, and this includes fighting skills, i.e., martial arts skills. Militaries mainly taught armed martial arts, such as archery, swordsmanship, spear fighting (on foot and on horseback). Sports would encourage wrestling, and in some places boxing and stickfighting. Policing would encourage armed martial arts with less-than-lethal weapons, and unarmed fighting.

There are two links between monks and martial arts. One is on an individual level, and the other is on an organisational (i.e., monastery) level. Large monasteries (e.g., in China and Japan, and also elsewhere) were major local landowners. They engaged in agriculture, trade, and industrial production, just like secular local landowners. Just like many major secular landowners, many monasteries would form their own private armies to defend their resources (e.g., organise their tenants into militias, recruit mercenaries, etc.) Monasteries could and did appoint soldiers as monks, giving them official status as part of the organisation as somebody other than a soldier/mercenary/thug, while not requiring any actual monk-like behaviour other than shaving the head. The martial arts involved in this military activity were the same as those of local landowners organising their militias - conventional military armed martial arts.

Once major monasteries had significant military forces, they tended to become involved in regional power struggles. Monasteries that sided with and provided military support for those who then proceed to win the civil wars of the time could gain significant support and patronage from the new rulers, while those who supported the wrong side could find their monasteries destroyed, either during the war, or afterwards. For example, the earliest known connection between Shaolin Monastery and martial arts dates to 621, when a local warlord siezed some land held by Shaolin. The monks raised an army and fought back; this action supported (deliberately? accidentally?) Li Shimin's capture of the city of Luoyang. Li Shimin became the Tang emperor Taizong, and Shaolin's ownership of the disputed land was confirmed, and monks were given official military titles. This also protected Shaolin from any seizures of their land by the Tang.

On an individual level, many monks, or bandits and drifters pretending to be monks, practiced martial arts. Itinerant monks travelled, and travel could be dangerous, and fighting skills could be useful. The soldiers making up the military forces of a particular monastery would travel, taking their skills with them. Former soldiers would become monks, bringing their fighting experience and skills with them. The world of the monk was not completely separate from the world of violence.

None of the above constitutes any religious connection with martial arts. It's about military force to maintain local power, and individual martial arts skills in connection with that, and with personal safety of travellers.

Some Chinese monasteries became famed for their martial arts in the late Ming. This was a time when a number of monasteries (including Shaolin) contributed forces to government armies to fight pirates, suppress rebellions,and so on. Where monastic forces were successful, they earned official praise, and gained fame. Coupled with the growth in popular fiction with the growth of printing, the link between martial arts and monks in China became firmly established. Since these monasteries were Buddhist, this led to anti-Manchu sentiment being expressed during the Qing in the form of Daoist martial arts (Daoist = "Chinese", Buddhist = "foreign"), with apparently Shaolin-derived martial arts being given a Daoist veneer and fabricated ancient histories.

The overlap between monasteries and violence did attract unfavourable comment in the religious world, and unfavourable attention from governments. Much of this was due to the purely nominal monkhood of many of those who made up the fighting forces - violent, meat-eating, uneducated, thieving men who, in the eyes of many, shaved their heads purely to escape punishment for their crimes (and in the eyes of most, did not make a positive religious contribution or example). That many, or most, fighting "monks" were not proper monks was well known.

Some further reading:

Adolphson, Mikael S. (2007). The Teeth and Claws of the Buddha. Monastic Warriors and Sōhei in Japanese History. Honolulu: University of Hawai‘i Press.

Peter Lorge, Chinese Martial Arts: From Antiquity to the Twenty-First Century: https://books.google.com/books?hl=en&lr=&id=Nzjv0uPqUyMC

Shahar, Meir. The Shaolin Monastery: History, Religion, and the Chinese Martial Arts. Honolulu: University of Hawai'i Press, 2008